No, he does not
The free intrusion happens every time a player rolls a 1 on the d20, rerolling would allow you to try succeeding once more, while not rerolling would not only trigger the GM Intrusion but also fail the test. The "free" gm intrusion is one of the special results of a check:
If a character rolls a natural 1, 17, 18, 19, or 20
(meaning the d20 shows that number), special rules come into play. These are explained in more detail in the following sections.
1: Intrusion. The GM makes a free intrusion (see below) and doesn’t award experience points (XP) for it.
17: Damage Bonus. If the roll was an attack, it deals 1 additional point of damage.
18: Damage Bonus. If the roll was an attack, it deals 2 additional points of damage.
19: Minor Effect. If the roll was an attack, it deals 3 additional points of damage. If the roll was something other than an attack, the PC gets a minor effect in addition to the normal results of the task.
20: Major Effect. If the roll was an attack, it deals 4 additional points of damage. If the roll was something other than an attack, the PC gets a major effect in addition to the normal results of the task. If the PC spent points from a stat Pool on the action, the point cost for the action decreases to 0, meaning the character regains those points as if she had not spent them at all.
And when you spend an experience point to reroll a check, you take the best of the two results. If you took the best result, your result was never 1 (unless you roll another 1), and thus the GM cannot make a gm intrusion for free:
The easiest, most straightforward way for a player to use XP is to reroll any roll in the game—even one that she didn’t make herself. This costs 1 XP per reroll. The player can reroll and use either the new roll or the original, whichever is better. She can continue to spend XP on more rerolls, but this can quickly become an expensive proposition. It’s a fine way to attempt to prevent a disaster, but it’s not a good idea to use of a lot of XP to reroll a single action over and over.
As mentioned earlier, a player can also spend 1 XP to refuse a GM intrusion.
If ruled otherwise, you could also be in a situation where it is possible to roll 17-20 two times and take both special roll results. If you have a lot of XP to spend, I don't see how that isn't always a good deal, as you still get to keep one good result.